Deriving the Equation of a Parabola Given a Focus and Directrix In this task, you will create a vertical parabola and a horizontal parabola based on the instructions provided. You will also practice writing equations of these parabolas.
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Question 1
The vertex form of the equation of a vertical parabola is given by , where (h, k) is the vertex of the parabola and the absolute value of p is the distance from the vertex to the focus, which is also the distance from the vertex to the directrix. You will use the GeoGebra geometry tool to create a vertical parabola and write the vertex form of its equation. Open GeoGebra, and complete each step below. If you need help, follow these instructions for using GeoGebra.

Part A
Mark the focus of the parabola you are going to create at F(6, 4). Draw a horizontal line that is 6 units below the focus. This line will be the directrix of your parabola. What is the equation of the line?

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Part B
Construct the line that is perpendicular to the directrix and passes through the focus. This line will be the axis of symmetry of the parabola. What are the coordinates of the point of intersection, A, of the axis of symmetry and the directrix of the parabola?

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Part C
Explain how you can locate the vertex, V, of the parabola with the given focus and directrix. Write the coordinates of the vertex.

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Part D
Which way will the parabola open? Explain.

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Part E
How can you find the value of p? Is the value of p for your parabola positive or negative? Explain.

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Part F
What is the value of p for your parabola?

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Part G
Based on your responses to parts C and E above, write the equation of the parabola in vertex form. Show your work.

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Part H
Construct the parabola using the parabola tool in GeoGebra. Take a screenshot of your work, save it, and insert the image below.

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Part I
Once you have constructed the parabola, use GeoGebra to display its equation. In the space below, rearrange the equation of the parabola shown in GeoGebra, and check whether it matches the equation in the vertex form that you wrote in part G. Show your work.

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Part J
To practice writing the equations of vertical parabolas, write the equations of these parabolas in vertex form:

focus at (-5, -3), and directrix y = -6
focus at (10, -4), and directrix y = 6.

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Question 2
The vertex form of the equation of a horizontal parabola is given by , where (h, k) is the vertex of the parabola and the absolute value of p is the distance from the vertex to the focus, which is also the distance from the vertex to the directrix. You will use GeoGebra to create a horizontal parabola and write the vertex form of its equation. Open GeoGebra, and complete each step below.

Part A
Mark the focus of the parabola you are going to create at F(-5, 2). Draw a vertical line that is 8 units to the right of the focus. This line will be the directrix of your parabola. What is the equation of the line?

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Part B
Construct the line that is perpendicular to the directrix and passes through the focus. This line will be the axis of symmetry of the parabola. What are the coordinates of the point of intersection, A, of the axis of symmetry and the directrix of the parabola?

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Part C
Explain how you can locate the vertex, V, of the parabola with the given focus and directrix. Write the coordinates of the vertex.

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Part D
Which way will the parabola open? Explain.

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Part E
How can you find the value of p? Is the value of p for your parabola positive or negative? Explain.

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Part F
What is the value of p for your parabola?

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Part G
Based on your responses to parts C and E above, write the equation of the parabola in vertex form. Show your work.

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Part H
Construct the parabola using the parabola tool in GeoGebra. Take a screenshot of your work, save it, and insert the image below.

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Part I
Once you have constructed the parabola, use GeoGebra to display its equation. In the space below, rearrange the equation of the parabola shown in GeoGebra, and check whether it matches the equation in the vertex form that you wrote in part G. Show your work.

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Part J
To practice writing the equations of horizontal parabolas, write the equations of these two parabolas in vertex form:

focus at (4, 3), and directrix x = 2
focus at (2, -1), and directrix x = 8